A285898 Triangle read by row: T(n,k) = number of partitions of n into exactly k consecutive parts (1 <= k <= n).

1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Author

Omar E. Pol and N. J. A. Sloane, Apr 28 2017

Keywords

• nonn
• tabl
• easy

Comments

To partition n into k parts, we see if m exists such that m + (m + 1) + ... + (m + k - 1) = k*m + binomial(k, 2) = n exists. a(n) = 1 if and only if (n - binomial(k, 2)) / k is an integer and larger than 0. - David A. Corneth, Apr 28 2017

It appears that this a full version of the irregular triangle A237048. - Omar E. Pol, Apr 28 2017

The value of a(n) can never exceed 1, since that would imply the existence of distinct equal-length ranges of consecutive integers that add up to the same number, which is impossible. - Sidney Cadot, Jan 22 2023

Examples

			Triangle begins:
1;
1, 0;
1, 1, 0;
1, 0, 0, 0;
1, 1, 0, 0, 0;
1, 0, 1, 0, 0, 0;
1, 1, 0, 0, 0, 0, 0;
1, 0, 0, 0, 0, 0, 0, 0;
1, 1, 1, 0, 0, 0, 0, 0, 0;
1, 0, 0, 1, 0, 0, 0, 0, 0, 0;
1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0;
1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0;
1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
...
For n = 15 there are four partitions of 15 into consecutive parts: [15], [8, 7], [6, 5, 4] and [5, 4, 3, 2, 1]. These partitions are formed by 1, 2, 3 and 5 consecutive parts respectively, so the 15th row of the triangle is [1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0].
		

Crossrefs

Row sums give A001227.

Cf. A000203, A038547, A067742, A082647, A131576, A204217, A237048, A237593, A245579, A261699, A279387, A281009, A204217.

Programs

Maple

A285898 := proc(n)
    corn := (n-binomial(k,2))/k ;
    if type(corn,'integer') then
        if corn > 0 then
            1 ;
        else
            0;
        end if;
    else
        0 ;
    end if;
end proc: # R. J. Mathar, Apr 30 2017

Mathematica

Table[Function[t, Function[s, ReplacePart[s, Map[# -> 1 &, t]]]@ ConstantArray[0, n]]@ Map[Length, Select[IntegerPartitions@ n, Length@ # == 1 || Union@ Differences@ # == {-1} &]], {n, 15}] // Flatten (* Michael De Vlieger, Apr 28 2017 *)

PARI

T(n, k) = n-=binomial(k, 2); if(n>0,n%k==0) \\ David A. Corneth, Apr 28 2017

Python

from sympy import binomial
def T(n, k):
    n=n - binomial(k, 2)
    if n>0:
        return 1 if n%k==0 else 0
    return 0
for n in range(1, 21): print([T(n, k) for k in range(1, n + 1)]) # Indranil Ghosh, Apr 28 2017

Formula

A000203(n) = Sum_{k=1..n} (-1)^(k-1) * ((Sum_{j=k..n} T(j,k))^2 - (Sum_{j=k..n} T(j-1,k))^2), assuming that T(k-1,k) = 0. - Omar E. Pol, Oct 10 2018